Do I have to stay in a faction the entire time an item is being produced, or can I leave the faction the moment I start the item?

You only need to be in the faction to start production. It won't stop if you leave.

However, you will want to ensure thta the operator of the facility/station you are producing in is set to a member of the managing faction or production will pause just before it completes.

Is that a bug or a feature?

From what I observed the operator was automatically changed to the leader of the managing production faction when it was finishing,

Xand, that sounds more like the leader or someone doing that rather than the system. Unless its been changed (and I know when Sin was discussing the re-code of production he liked the idea of not having a check at the end of production since that was a cause of many bugs) its used to check at the end of production if all the requirements are met. If the operator was incorrect, production was halted at 99% until an appropiate operator was assigned and production was restarted.

That would explain why it's on the rules page. =p

It definitely changes the operator to the faction leader if the operator isn't in the faction, I'm not sure whether it's supposed to pause or not. (or whether it even pauses in the first place)